A Guided Walk Down Wall Street: An Introduction to Econophysics

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1060 Giovani L. Vasconcelos where r = 1 T T∑ r(t ′ ). (144) t ′ =1 Next we break up X(t) into N non-overlapping time intervals, I n , of equal size τ, where n = 0, 1, ..., N − 1 and N corresponds to the integer part of T/τ. We then introduce the local trend function Y τ (t) defined by Y τ (t) = a n + b n t for t ∈ I n , (145) where the coefficients a n and b n represent the least-square linear fit of X(t) in the interval I n . Finally, we compute the rescaled fluctuation function F (τ) defined as [35] F (τ) = 1 √ 1 ∑Nτ [X(t) − Y τ (t)] 2 , (146) S nτ t=1 where S is the data standard deviation S = √ 1 T∑ (r t − r) 2 . (147) T t=1 The Hurst exponent H is then obtained from the scaling behavior of F (τ): F (τ) = Cτ H , (148) where C is a constant independent of the time lag τ. In a double-logarithmic plot the relationship (148) yields a straight line whose slope is precisely the exponent H, and so a linear regression of the empirical F (τ) will immediately give H. One practical problem with this method, however, is that the values obtained for H are somewhat dependent on the choice of the interval within which to perform the linear fit [35, 36]. It is possible to avoid part of this difficulty by relying on the fact that for the fractional Brownian motion, the fluctuation function F (τ) can be computed exactly [31]: F H (τ) = C H τ H , (149) where [ 2 C H = 2H + 1 + 1 H + 2 − 2 ] 1/2 . (150) H + 1 In (149) we have added a subscript H to the function F to denote explicitly that it refers to W H (t). Equation (149) with (150) now gives a one-parameter estimator for the exponent H: one has simply to adjust H so as to obtain the best agreement between the theoretical curve predicted by F H (τ) and the empirical data for F (τ). 7.3 Fractional Brownian motion in Finance The idea of using the FMB for modeling asset price dynamics dates back to the work of Mandelbrot & van Ness [37]. Since then, the Hurst exponent has been calculated (using different estimators) for many financial time series, such as stock prices, stock indexes and currency exchange rates [38, 39, 40, 41, 35]. In many cases [38] an exponent H > 1/2 has been found, indicating the existence of longrange correlation (persistence) in the data. It is to be noted, however, that the values of H computed using the traditional R/S-analysis, such as those quoted in [38], should be viewed with some caution, for this method has been shown [35] to overestimate the value of H. In this sense, the DFA appears to give a more reliable estimates for H. An example of the DFA applied to the returns of the Ibovespa stock index is shown in Fig. 10 (upper curve). In this figure the upper straight line corresponds to the theoretical curve F H (τ) given in (149) with H = 0.6, and one sees an excellent agreement with the empirical data up to τ ≃ 130 days. The fact that H > 0.5 thus indicates persistence in the Ibovespa returns. For τ > 130 the data deviate from the initial scaling behavior and cross over to a regime with a slope closer to 1/2, meaning that the Ibovespa looses its ‘memory’ after a period of about 6 months. Also shown in Fig. 10 is the corresponding F (τ) calculated for the shuffled Ibovespa returns. In this case we obtain an almost perfect scaling with H = 1/2, as expected, since the shuffling procedure tends to destroys any previously existing correlation. F(τ) 10 1 10 0 ibovespa shuffled data H=0.6 H=0.5 10 1 10 2 10 3 10 4 τ Figure 10. Fluctuation function F (τ) as a function of τ for the returns of the Ibovespa index (upper curve) and for the shuffled data (lower curve). The upper (lower) straight line gives the theoretical curve F H(τ) for H = 0.6 (H = 1/2). As already mentioned, the Hurst exponent has been calculated for many financial time series. In the case of stock indexes, the following interesting picture appears to be emerging from recent studies [40, 41, 35]: large and more developed markets, such as the New York and the London Stock Exchanges, usually have H equal to (or slightly less than) 1/2, whereas less developed markets show a tendency to have H > 1/2. In other words, large markets seem indeed to be ‘efficient’ in the sense that H ≃ 1/2, whereas less developed markets tend to exhibit long-range correlation. A possible interpretation for this finding is that smaller markets are conceivably more prone to ‘correlated fluctuations’ and perhaps more susceptible to being pushed around by aggressive investors, which may explain in part a Hurst exponent greater than 1/2.
Brazilian Journal of Physics, vol. 34, no. 3B, September, 2004 1061 It should also be pointed out that a considerable timevariability of the exponent H for stock indexes has been found [41, 35], indicating that the data in such cases cannot be modeled in terms of stationary stochastic processes. (A time-varying H has also been observed in other financial data, such as, currency exchange rate [39].) In such cases, the stationarity assumption is only a rather crude approximation [35]. Furthermore, the time dependence of the Hurst exponent is an indication that the underlying process might be multifractal rather than monofractal; see below. H 0.9 0.8 0.7 0.6 0.5 0.4 0.3 1968 1972 1976 1980 1984 1988 1992 1996 2000 t Figure 11. The Hurst exponent H for the Ibovespa as a function of time. Here H was computed in two-year time intervals, with the variable t denoting the origin of each such interval. A time-varying Hurst exponent has been observed for the Brazilian stock market. This case is of particular interest because in the recent past Brazil was plagued by runaway inflation and endured several ill-fated economic plans designed to control it. To analyze the effect of inflation and the economic plans, Costa and Vasconcelos [35] calculated a time-varying Hurst exponent for the Ibovespa returns, computed in three-year time windows for the period 1968–2001. A similar analysis but with two-year time windows is shown in Fig. 11. One sees from this figure that during the 1970’s and 1980’s the curve H(t) stays well above 1/2, the only exception to this trend occurring around the year 1986 when H dips momentarily towards 1/2—an effect caused by the launch of the Cruzado economic Plan in February 1986 [35]. In the early 1990’s, after the launching of the Collor Plan, we observe a dramatic decline in the curve H(t) towards 1/2, after which it remained (within some fluctuation) around 1/2. This fact has led Costa and Vasconcelos [35] to conclude that the opening and consequent modernization of the Brazilian economy that begun with the Collor Plan resulted in a more efficient stock market, in the sense that H ≃ 0.5 after 1990. In Fig. 11, one clearly sees that after the launching of a new major economic plan, such as the Cruzado Plan in 1986 and the Collor Plan in 1990, the Hurst exponent decreases. This effect has, of course, a simple economic interpretation: a Government intervention on the market is usually designed to introduce “anti-persistent effects,” which in turn leads to a momentary reduction of H. Note also that only after 1990 does the curves H(t) goes below 1/2. This finding confirms the scenario described above that more developed markets (as Brazil became after 1990) tend to have H < ∼ 1/2. 7.4 Option pricing under the FBM assumption We have seen above that often times asset prices have a Hurst exponent different from 1/2. In such cases, the standard Black-Scholes model does not apply, since it assumes that returns follow a Brownian motion (H = 0.5). A more appropriate model for the return dynamics would be the fractional Brownian motion (FBM). Indeed, a ‘fractional Black-Scholes model’ has been formulated, in which it is assumed that the stock price S follows a geometric fractional Brownian motion given by dS = µSdt + σSdW H , (151) where W H (t) is the standard FBM. The fractional stochastic differential equation above is shorthand for the integral equation S(t) = S(0)+µ ∫ t 0 S(t ′ )dt ′ +σ ∫ t 0 S(t ′ )dW H (t ′ ). (152) To make mathematical sense of this equation is, of course, necessary to define stochastic integrals with respect to W H (t). Here it suffices to say that a fractional Itô calculus can indeed be rigorously defined [42] that shares (in an appropriate sense) many of the properties of the usual Itô calculus. In the context of this fractional Itô calculus is possible to prove that the solution to (151) is given by { S(t) = S(0) exp µt − 1 } 2 σ2 t 2H + σW H (t) . (153) Compare this expression with (53). One can show [42] that the fractional Black-Scholes model is complete and arbitrage-free. To price derivatives with this model, one can apply the same ∆-hedging argument used before, which now leads to the ‘fractional Black- Scholes equation.’ The result is summarized in the following theorem. Theorem 4 In the fractional Black-Scholes model, the price F (S, t) of a European contingent claim with payoff Φ(S(T )) is given by the solution to the following boundaryvalue problem ∂F ∂t + Hσ2 t 2H−2 S 2 ∂2 F ∂F + rS − rF = 0, ∂S2 ∂S (154) F (T, S) = Φ(S). (155)
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A Guided Walk Down Wall Street: An Introduction to Econophysics

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